The Eutrigon Theorem

The black central triangle with one angle equal to 60° is called a eutrigon. The areas of equilateral triangles constructed on the three faces , , and obey the eutrigon theorem, which gives the area of the black triangle in terms of the areas of the other three triangles.

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Here is a quick proof: The area of an equilateral triangle with side is . By the law of sines, the area of a eutrigon is . The law of cosines gives , because . Multiplying by gives the statement of the theorem.

Snapshot 1: when , the figure reduces to four equal equilateral triangles. The validity of the theorem then becomes trivial

Snapshot 2: when , the eutrigon becomes a 30-60-90 right triangle. By Pythagoras' theorem, , implying that the black triangle is twice the area of the blue triangle

Snapshot 3: a degenerate case, with , which collapses the eutrigon

Reference: "The Eutrigon Theorem - a new* twin to the theorem of Pythagoras" on this website.